We explore the effect of the dimensionless numbers and flow parameters that characterise the system (see Eq. (REF )) considering the following ranges: \(Re = 100 - 443\) , \(Ca = 0.0089 - 0.0693\)  (\(We = 3.94 - 30.70\) , \(Oh = 4.48\times 10^{-3} - 1.25\times 10^{-2}\) ), \(Pe_{c,s} = 100\) , \(Bi = 0.10 - 1\) , \(Da = 0.01 - 1\) , \(k = 0.10 - 10.00\) , and \(\beta _{s} = 0.25 - 1\) . For all cases simulated, the viscosity and density ratio between the phases is kept constant and in line with representative values for water and air: \(\rho _{l}/\rho _{g}\sim \)  O(\(10^{3}\) ) and \(\mu _{l}/\mu _{g}\sim \)  O(\(10^{2}\) ). We define a `base' case from which our parametric sweep was carried out with the following conditions: \(Re = 443\) , \(Ca = 0.0693\) , \(Pe_{c,s} = 100\) , \(Bi = 0.10\) , \(Da = 1\) , \(k = 0.10\) , and \(\beta _{s} = 0.50\) . Unless stated otherwise, all results correspond to simulations under these conditions. The selection of the testing ranges was based on common values encountered in bubbly flow systems in the presence of surfactants, as reported in [1]}, as well as values of \(We\)  that would allow a significant disruption of the bubble back to generate the aforementioned re-entrant cavity and consequent drop encapsulation [2]}, [3]}. For typical ionic and non-ionic surfactants in water, such as sodium dodecyl sulphate (SDS), N-dodecyl-N,N-dimethylammonio-3-propane sulfonate, and Triton X-100 (TX100), the value of \(\Gamma _{\infty }\)  ranges between O(\(10^{-6}-10^{-5}\) ) mol/m\(^{2}\)  and \(D_{s}\)  between O(\(10^{-12}-10^{-8}\) ) m\(^{2}/\) s, which results in \(Pe_{s}=\textit {O}(10^{3}-10^{6})\)  [4]}, [5]}. As noted previously in [6]} and [7]}, saturation of the system's dynamics is reached above \(Pe_{s} \sim 100\) . \(Pe_{c}\)  was set to be equal to \(Pe_{s}\) , following the study of [8]}.
